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At the height of the galactic empire how do think they should go about conquering the unknown regions?

Also, getting the other large governments to submit to the emperor's rule as well?

I think it is difficult to make gains in the unknown regions due to presence of a large number of independent and/or loosely-controlled systems. Two many governments, factions and systems to worry about.

Republic, in contrast, is a straightforward and identifiable target.

i probably would have sent agents to make deals with the smaller governments, asking them to submit to the emperor in exchange for wealth and the territories of the governments that refused to submit. then i would use their forces to help the empire's conquer those that wouldnt submit, but unstead of using actual imperial forces to attack i would use thrawn's empire of the hand to do it. but in using the smaller governments to wipe out the others, you effectively get rid of potential threats both from your allies and your enemies.

i also might do what hitler wanted to do and that was to ue counterfeit money to cause inflation in the other nations and make them go bankrupt and cause chaos. this might work for the bigger governments like the chiss.

plus the empire was already on good terms with the ssr'ruvi imperium so they could work something out there.

you could use assassinations from force-sensitives, so as to not leave a trace, bribes, anything you wanted. just as long as you didnt mess with too many people at once so you get rid of a couple of threats at once instead of having to fight alot of people at once.

Build a Death Star